Clones from comonoids

TL;DR

This paper investigates the conditions under which the endomorphism operad of a comonoid forms a clone in braided monoidal categories, revealing that cocommutativity is not guaranteed.

Contribution

It provides an explicit example showing that the endomorphism operad being a clone does not imply cocommutativity of the comonoid.

Findings

Endomorphism operad of a comonoid need not be a clone in braided categories.

Cocommutativity of a comonoid is not implied by its endomorphism operad being a clone.

Explicit example demonstrating the non-equivalence of these properties.

Abstract

The fact that the cocommutative comonoids in a symmetric monoidal category form the best possible approximation by a cartesian category is revisited when the original category is only braided monoidal. This leads to the question when the endomorphism operad of a comonoid is a clone (a Lawvere theory). By giving an explicit example, we prove that this does not imply that the comonoid is cocommutative.